# Prove matrix $A=(\cos\langle\alpha_i,\epsilon_j\rangle)_{n\times n}$ is orthogonal

Assume $$\{\alpha_1,\cdots\alpha_n\},\{\epsilon_1,\cdots,\epsilon_n\}$$ are both orthonormal basis of Euclidean Space $$V$$. Consider the matrix $$A=(\cos\langle\alpha_i,\epsilon_j\rangle)_{n\times n}$$ $$\langle\alpha_i,\epsilon_j\rangle$$ denotes the angle between $$\alpha_i,\epsilon_j$$.
Prove 1. A is orthogonal. 2. Every orthogonal matrix can express as the form of $$A$$.

i.e prove $$\cos\langle\alpha_i,\epsilon_1\rangle\cdot \cos\langle\alpha_j,\epsilon_1\rangle+\cos\langle\alpha_i,\epsilon_2\rangle\cdot\cos\langle\alpha_j,\epsilon_2\rangle+\\ \cdots+\cos\langle\alpha_i,\epsilon_n\rangle\cdot\cos\langle\alpha_j,\epsilon_n\rangle=0$$ for $$i\ne j$$.

And

$$\cos\langle\alpha_i,\epsilon_1\rangle\cdot \cos\langle\alpha_j,\epsilon_1\rangle+\cos\langle\alpha_i,\epsilon_2\rangle\cdot\cos\langle\alpha_j,\epsilon_2\rangle+\\ \cdots+\cos\langle\alpha_i,\epsilon_n\rangle\cdot\cos\langle\alpha_j,\epsilon_n\rangle=1$$ for $$i=j$$.

How to deal with the sum?

• So... I guess $\cos<\alpha_i,\epsilon_j>$ is just a very convoluted way to express the usual meaning of $\langle \alpha_i,\epsilon_j\rangle$? – Saucy O'Path Oct 18 '18 at 7:23
• @ Saucy O'Path It's the cosine of the angle between $\alpha_i,\epsilon_j$ – Jaqen Chou Oct 18 '18 at 7:25
• Yes, "a very convoluted way..." et cetera. – Saucy O'Path Oct 18 '18 at 7:26
• huh... what's the usual way... – Jaqen Chou Oct 18 '18 at 7:34

Hint. The cosine of the angle between two unit vectors $$u$$ and $$v$$ is just their dot product, i.e. $$u^Tv$$. So, if $$P$$ is the augmented matrix containing the $$\alpha_j$$s as columns and $$Q$$ is the augmented matrix containing the $$\varepsilon_j$$s as columns, then $$A=P^TQ$$.
For the second part, take $$Q=A$$ with an appropriate $$P$$.
• So since $P,Q$ are orthogonal,$A^TA=Q^TPP^TQ=I$. Could you also give some hints about every orthogonal have form of this $A=P^TQ$? – Jaqen Chou Oct 18 '18 at 9:21
• emm... I am trying to find out exactly $P$ and $Q$..Am I miss something... – Jaqen Chou Oct 18 '18 at 9:47